Logical Axioms

Quantifier Axioms

• Suppose $\phi$ and $\psi$ are formulas and $v$ is a variable not free in $\phi$ $\forall v(\phi \to \psi )\to (\phi \to \forall v\psi )$
• Suppose $\phi$(written $\phi (v)$ here, but it may have other free variables) a formula of $\mathcal{L}$. Suppose $\phi (t)$ is obtained from $\phi$ by substituting the term $t$ for each free occurrence of $v$ in $\phi$, and that no variable of t occurs bound in φ at the places where it is substituted $\forall v\phi (v)\to \phi (t)$ (that is to say provided $t$ is “substitutable” for $v$ in $\phi (v)$)

eg: $\forall x(x=y\to x=y)\to (x=y\to \forall x(x=y))$ is not logically true

Universal Validity of the Logical Axioms

The logical axioms are universally valid.